Justification of the Formula for the
Sum of the Interior Angles of a Convex Polygon
Materials: 3" × 5" or similar size file card, 1 per person. Also rulers, letter size sheets of
blank paper, and scissors.
1. Use a ruler to draw an arbitrary triangle (different from others at your table) on a
file card. The base of the triangle should be between 1 and 3 inches and the triangle
should be scalene (no two sides with the same length). Cut out the triangle and
discard the rest of the card. Inside the triangle you just cut out, label the base with
the letter b. Draw a large T inside the triangle to ensure that you don't flip it over.
(The T must always be visible.)
2. Trace the triangle in the middle of a blank 8.5" × 11" sheet of paper with the base
parallel to the horizontal 11" side. Trace another copy next to it so that the two copies
share another complete side (not the base). (The T must always be visible.)
3. The bases of the two triangles are clearly congruent because they were traced from
the same triangle. By thinking carefully about how you transformed the stiff triangle
you used to trace, explain why the bases must be parallel.
Hint: What is the simplest one or two transformations you could use to move the
first triangle into the second. Specify the transformation(s) as precisely as you can.
4. Continue to trace copies of your triangle on the paper, always sharing a complete
side (which now could be the base) with one that’s already drawn, and always
keeping the T visible. Trace enough triangles so that the original triangle is
completely surrounded in all directions with a border of other triangles. These
triangles make a tiling or tessellation.
5. Discuss in your group what your tessellation tells you about the sum of the angles in
your triangle. Be prepared to convince others of this fact. Does this prove that the
sum of the interior angles of any triangle is 180˚?
6. Compare this justification with cutting out all the angles and placing them next to
each other. Are these both proofs?
PRIME2 Summer Institute 2013
7. Now we go on to polygons. The polygon shown below is concave. What makes it
concave? What confusion can arise when considering angles of a concave polygon?
Because of this confusion, we’ll stick with convex (non-concave) polygons for now.
8. On a fresh sheet of blank paper, draw one large convex polygon with the number of
sides assigned by your teacher. (Wait until you are assigned a number of sides before
drawing.) Those in your group with the same number of sides should work together.
The formula you found using pattern blocks for the sum of the interior angles of a
convex polygon has the number 180 in it. Using this fact as a hint, draw some
dashed segments in your diagram that you can use to figure out the sum of the
interior angles. Try to use a different method from your partner(s). Make a smaller
copy of your diagram below.
9. Compare the different methods used at your table. Does the result agree with the
formula you found using Pattern Blocks? Are these methods different proofs of the
formula?
PRIME2 Summer Institute 2013
Extensions
1. Find a formula for the measure of each interior angle of a regular polygon with n
sides.
2. Can you interpret this formula so that it applies to concave polygons as well as
convex ones? Try it!
3. Which regular polygons can be made with pattern blocks? (OK to use lots of them.)
4. How do you know that you have made all possible regular polygons with the pattern
blocks?
PRIME2 Summer Institute 2013